Theorem rfovfvd 38798

Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, and relation 𝑅. (Contributed by RP, 25-Apr-2021.)

Hypotheses

Ref	Expression
rfovd.rf	⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))
rfovd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rfovd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
rfovfvd.r	⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
rfovfvd.f	⊢ 𝐹 = (𝐴𝑂𝐵)

Assertion

Ref	Expression
rfovfvd	⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑟,𝑥   𝐵,𝑎,𝑏,𝑟,𝑥   𝑦,𝐵,𝑎,𝑏,𝑟   𝑅,𝑟,𝑥   𝑦,𝑅   𝜑,𝑎,𝑏,𝑟

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝑅(𝑎,𝑏)   𝐹(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑟,𝑎,𝑏)

Proof of Theorem rfovfvd

Step	Hyp	Ref	Expression
1		rfovfvd.f	. . 3 ⊢ 𝐹 = (𝐴𝑂𝐵)
2		rfovd.rf	. . . 4 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))
3		rfovd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		rfovd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5	2, 3, 4	rfovd 38797	. . 3 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
6	1, 5	syl5eq 2806	. 2 ⊢ (𝜑 → 𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
7		breq 4806	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
8	7	rabbidv 3329	. . . 4 ⊢ (𝑟 = 𝑅 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦})
9	8	mpteq2dv 4897	. . 3 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))
10	9	adantl 473	. 2 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))
11		rfovfvd.r	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
12		mptexg 6648	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}) ∈ V)
13	3, 12	syl 17	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}) ∈ V)
14	6, 10, 11, 13	fvmptd 6450	1 ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  {crab 3054  Vcvv 3340  𝒫 cpw 4302   class class class wbr 4804   ↦ cmpt 4881   × cxp 5264  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818

This theorem is referenced by:  rfovfvfvd  38799